PhD Thesis in
Civil and Environmental Engineering
Cycle XXVI
Modelling and analysis of
thin-walled beams in the context
of the Generalized Beam Theory
PhD Candidate:
Alejandro Gutierrez
PhD coordinator:
Prof. Alberto Lamberti
Advisor: Prof. Stefano de Miranda
Co-Advisor: Prof. Francesco Ubertini
Tesi di Dottorato di Ricerca in
Ingegneria Civile e Ambientale
XXVI Ciclo
Settore Scientifico-Disciplinare ICAR/08
Settore Concorsuale 08/B2
Modellazione e analisi di travi in
parete sottile nell’ambito della
Generalized Beam Theory
Candidato:
Alejandro Gutierrez
Coordinatore:
Prof. Alberto Lamberti
Relatore: Prof. Stefano de Miranda
Correlatore: Prof. Francesco Ubertini
Introduction xv
1 An overview of the mechanics of thin-walled
mem-bers 1
1.1 Cold-formed profiles . . . 4
1.2 Composite profiles . . . 7
1.3 Theoretical framework . . . 12
1.3.1 The theory of Capurso . . . 14
1.3.2 The Generalized Beam Theory . . . 16
1.4 Numerical Modelling . . . 22
1.4.1 The Finite Strip Method . . . 25
2 A Generalized Beam Theory with shear deforma-tion 29 2.1 New kinematics . . . 33
2.2 Strains and generalized deformations . . . 34
2.3 Generalized stresses . . . 37
2.4 Cross section stiffness matrix . . . 38
2.5 Deformation modes . . . 41
2.5.2 Shear modes . . . 43 2.6 Cross section analysis . . . 45 2.6.1 Modal decomposition of flexural modes . . . 46 2.6.2 Rigid-body modes . . . 48 2.6.3 Modal decomposition of basic shear modes . 50 2.6.4 Modal decomposition of additional shear modes 51 2.7 Cross-section stiffness matrix and generalized strain
parameters in the modal space . . . 52 2.8 A GBT-based finite element . . . 58 2.9 Some examples . . . 60 2.9.1 Test 1: C-section cantilever beam under torsion 60 2.9.2 Test 2: Z-section cantilever beam under
transver-sal load . . . 61 2.9.3 Test 3: C-section fixed-fixed beam under a
transversal load. . . 63
3 Stress reconstruction in the framework of the GBT 67 3.1 Reconstruction of the three-dimensional stresses . . . 69 3.2 Recovery of the generalized stresses . . . 77 3.3 Some examples . . . 79
4 Constitutive relations: from the isotropic to the
com-posite beam 91
4.1 The constitutive relation in the conventional GBT . 92 4.1.1 The isotropic beam . . . 92 4.1.2 The composite beam . . . 95
4.2 An alternative approach to obtain the constitutive
law of the beam model . . . 106
4.2.1 The laminated case . . . 111
4.3 Shear correction factors . . . 115
4.4 Some examples . . . 118
4.4.1 An isotropic beam . . . 119
4.4.2 An orthotropic beam . . . 120
4.4.3 A laminated beam . . . 124
4.4.4 Shear correction factors . . . 127
1.1 Example of a thin-walled cross-section. . . 2 1.2 Some cold-formed sections. . . 5 1.3 Applications of cold-formed structural elements . . . 5 1.4 Diagram of the roll-forming process. . . 6 1.5 Schematic view of the hand laminating and autoclave
processes . . . 10 1.6 Schematic view of the filament winding and fiber
placing processes . . . 10 1.7 Schematic view of the pultrusion process . . . 11 1.8 Schematic view of the liquid molding process . . . . 12 1.9 Buckling modes . . . 13 1.10 Capurso theory: global and local reference systems
on a generic thin-walled cross-section. . . 15 1.11 GBT: natural and internal nodes in a generic
thin-walled cross-section. . . 19 1.12 GBT: (a) warping in a fundamental flexural mode,
(b) in-plane displacements in a fundamental flexural mode, (c) in-plane displacements in a local flexural
1.13 C cross-section modelled by GBT: in- and out-of-plane displacements corresponding to the six
funda-mental flexural modes after modal decomposition. . 23 1.14 Comparison of the level of discretization used for the
same geometry using finite elements and finite strips 25
2.1 Sketch of the main difference between classical shear deformable GBT and the present formulation: (a) undeformed elementary beam, (b) bending strain com-ponent, (c) corresponding shear strain component in the classical GBT, (d) corresponding shear strain component in the Timoshenko beam theory and in
the present theory. . . 32 2.2 Pattern of the cross-section stiffness matrix for nF =
nS = 6: (a) in the natural space, (b) in the modal
space, (c) in the modal space after reordering. . . 47 2.3 Schematic view of the effect of a flexural mode and
of the associated shear mode on a two-nodes
cross-section. . . 56 2.4 C-section cantilever beam under torsion. . . 61 2.5 Z-section cantilever beam under transversal load. . . 62 2.6 Z-section cantilever beam: y-displacement of node 4. 64 2.7 Z-section cantilever beam: y-displacement of node 2. 64 2.8 C-section fixed-fixed beam under transversal load. . 65 2.9 C-section fixed-fixed beam: y-displacement of the
3.1 Different assumptions made over the generic wall . . 72 3.2 Free body diagram of an arbitrary natural node. . . 76 3.3 Element patch used in the RCP recovery. . . 79 3.4 Z-section cantilever beam. . . 79 3.5 Z-section cantilever beam: in-plane displacements
associated to fundamental flexural modes (shear modes have null in-plane displacements). . . 80 3.6 Z-section cantilever beam: out-of-plane displacements
associated to fundamental flexural modes and shear
modes. . . 81 3.7 Normal stress σzz along the section midline at z/L =
0.5 and z/L = 0.75. . . 82 3.8 Normal stress σss along the section midline at z/L =
0.5 and z/L = 0.75. . . 83 3.9 Shear stress τzs along the section midline at z/L =
0.5 and z/L = 0.75. . . 84 3.10 Normal stress σzz along the section at z/L = 0.5 and
n = t/4. . . 85 3.11 Normal stress σss along the section at z/L = 0.5 and
n = t/4. . . 86 3.12 Shear stress τzs along the section z/L = 0.5 and
n = t/4. . . 87 3.13 Shear stress τsn along the section midline at z/L = 0.5. 88
3.15 Shear stresses along the wall thickness for z/L = 0.5: (a) τsn at midpoint of the lower flange; (b) τzn at
midpoint of the lower flange; (c) τsn at midpoint of
the web; (d) τzn at midpoint of the web; (e) τsn at
midpoint of the upper flange; (f) τzn at midpoint of
the upper flange. . . 89 3.16 Normal stress σnn along the wall thickness at the
midpoint of the upper flange for z/L = 0.5. . . 90
4.1 Orthotropic plate: material (x1, x2, x3) and local (n, s, z)
reference systems. . . 96 4.2 Reference system and layer numbering used for a
laminated plate. . . 105 4.3 Isotropic case: a hat-section cantilever beam under
transversal load. . . 119 4.4 Isotropic case: Normalized y-displacement of node 6. 120 4.5 Orthotropic case: a C-section cantilever beam under
transversal load. . . 121 4.6 Orthotropic case: normalized y-displacement of node
4, θ = 0. . . 122 4.7 Orthotropic case: normalized y-displacement of node
4, θ = π/2. . . 123 4.8 Orthotropic case: normalized y-displacement of node
4, θ = π/6. . . 124 4.9 Orthotropic case: normalized y-displacement of node
4.10 Laminated case: normalized y-displacement of node
4, stacking sequence [π/3; π/6; π/3]. . . 126 4.11 Laminated case: normalized y-displacement of node
4, stacking sequence [0; π/6; π/3]. . . 127 4.12 Shear correction factors: a Z-section cantilever beam
under transversal load. . . 128 4.13 Shear correction factors: Convergence rate of the
complementary in-plane membrane energy per unit
area. . . 129 4.14 Shear correction factors: effect of the shear
correc-tion factors κ1 and κ2 on the distribution of τzs over
En memoria del Catire Gutiérrez, la Vieja Candelaria, El Látigo Negro y El Loro, quienes se fueron del mundo sin verme hecho un hombre completo. Yo guardaré sus nombres y su memoria siempre con orgullo.
I would like to thank the European Commission for financing the first three years of my stay in Italy through the “Ánimo, ¡Chévere!” scholarship program. I thank also the Department of Civil and En-vironmental Engineering at the University of Bologna for financing the rest of my PhD studies.
Thanks to Professor Stefano de Miranda for his inexhaustible pa-tience and for the guidance he has provided in these years, not only on the matters dealt with in this work, but on the many commonly overlooked aspects of being a scientific researcher. Thanks also to Professor Francesco Ubertini for his advice and the trust he has given me on this endeavour. Also, I thank the gang at the depart-ment of Civil Engineering: Nicholas, Marco, Rosario, Alessandro, Ilaria, and the others. Without you guys, without the friendship and hospitality you have so freely given, this would have been much more difficult and considerably less fun!
I thank my old Venezuelan engineering comrades as well. Many of whom have, like myself, left behind our land looking for new horizons. Time and geography may have sundered our paths, but the bonds we once made will endure. Special thanks to Professor Carlos Graciano, who initiated me in scientific research.
Thanks, of course, to my old folks Blas and María. I don’t have enough words to express how much I value the unconditional love and support you have given me through my life. You are the best parents a man could ever dream of. I hope my deeds make you proud.
Thanks, finally, to my dear Bristin. You came into my life like a sudden spring after a long, forgotten winter. You give me joy, strength, courage and wisdom. You make me a better man in ways I never thought I could be.
Thin-walled beams can be broadly defined as slender structural el-ements with distinctive dimensions that are all of different orders of magnitude: their thickness is small when compared with the dimensions of the cross-section, which themselves are small when compared with the element length. This simple geometrical fact, apparently banal in itself, is in fact the source of the many advan-tages that distinguish thin-walled beams from other structural ele-ments: thin-walled beams are lightweight but can resist significant loads; they can be cheaper to produce, transport and install than more compact beams; they allow for versatility in material and shape, giving more freedom of choice to the structural designer. At the same time, the geometrical particularities of thin-walled beams may make their mechanical behaviour considerably complex: classical beam models often fall short of accurately describing the whole kinematic range of these elements, especially in the case of cold-formed or composite profiles where bending, torsion, cross-section distortion and local effects occur together and in a coupled manner. These complexities translate into a need for modelling and analysis tools that are both highly accurate and
computation-ally cheap. In this sense, the contributions to the modelling and analysis of thin-walled beams can be subdivided in two main cate-gories: one towards an effective three-dimensional "shell" modeling, and the other towards the formulation and development of mono-dimensional beam models with enriched cross-section kinematics. The present work stands within this second kind of approach.
In the first chapter of this work, an overview of the importance of thin-walled beams, their commercial forms, and their manufac-turing methods is outlined along with a brief description of some of the most important models used to analyze these elements. Firstly, cold-formed profiles are presented as one of the historically most common application of thin-walled beams. Indeed, the usage of these is ever increasing in the building business and the need to take into account the distortion of their cross section in their analysis is well documented in current standards [1]-[4]. Later on, composite profiles are introduced as a more recent innovation in the field, mo-tivated mainly by the needs of the aeronautical/aerospace industry, but also applied to civil, naval, and turbomachinery construction in the recent decades. After this brief introduction to the subject, some of the most relevant models formulated for thin-walled beams are introduced. The theory of Vlasov [5], the first approach to the modelling of a thin-walled beam, is based on a description of the cross-section warping related to a non-uniform distribution of tor-sional rotation. A detailed exposition of the Vlasov theory is found in the work of Bleich [6] and Timoshenko [7] with special
empha-sis on stability analyempha-sis. From this original model, considerable work has been done over the decades to enhance and extend the Vlasov theory by different methods, such as the incorporation of higher-order parameters in the displacement field. Some significant contributions to the ongoing efforts are those of Kang and Yoo [8], who developed a Vlasov-like model to study large displacement be-haviour in curved beams, and the work of Kim [9], Wilson [10], and Stavridis [11] among others who have studied the vibration and sta-bility analysis of thin-walled profiles. However, these formulations have maintained the basic Vlasov hypotheses of the cross-section being rigid in its own plane and of null shear deformability. Regard-ing the second of these, the work of Capurso [12], [13] extended the model of Vlasov to include shear deformation over the cross-section midline by generalizing the description of warping. More recently, the work of Piovan [14], Gendy and Saleeb [15], and again Kim [16] has revolved around the inclusion of shear deformability on Vlasov-like beam models for vibration and stability analysis. However, beam models based on the kinematics of Vlasov fail to take into account the effects of cross-section distortion and local in-plane de-formation of the walls. In this respect, The Generalized Beam The-ory (GBT), originally proposed by Schard [17], [18] in the 1980s, is a model that adds to the kinematics of Vlasov the distortion of the cross-section by means of expressing the deformation of the member as the superposition of a series of cross-sectional modes. Following the work of Schardt, many authors have contributed to the
improve-ment of the GBT by extending it beyond its original formulation for open unbranched cross-sections [19]-[21] and by adding nonlin-ear effects for the analysis of buckling problems [22]-[24]. Moreover, in the spirit of the Kantorovich’s semi-variational method, a new approach for the selection of the cross-section deformation modes has been presented in [25]. Recently, the application of the GBT to analyze cold-formed roof systems has been presented in [26]-[28].
As regards the shear deformation, Silvestre and Camotim [29]-[30] were the first to remove the Vlasov constraint of null shear deformation in the GBT. However, their formulation considered constant warping displacement over the wall thickness, which leads to null shear strain between the direction of the beam axis and that orthogonal to the wall midline. This in turn means that classical shear deformable beam theories are not recovered exactly, which limits the attractiveness of the resultant theory. In the second chap-ter of this work, following the ideas presented in [31],a new formu-lation of the GBT that coherently accounts for shear deformation is presented. In particular, a modified formulation of the kinemat-ics early proposed by Silvestre and Camotim for shear deformable GBT is devised. The new formulation introduces the shear de-formation along the wall thickness direction besides that along the wall midline, so guaranteeing a coherent matching between bending and shear strain components of the beam. According to the new kinematics, a reviewed form of the cross-section analysis procedure is devised, based on a unique modal decomposition for both flexural
and shear modes. Much attention is posed on the mechanical inter-pretation of the deformation parameters in the modal space. It is shown that, in the modal space, it is possible to clearly distinguish bending deflections from deflections due to shearing strains, and to recover classical beam degrees of freedom and standard beam the-ories as special cases. The effectiveness of the proposed approach is illustrated on typical benchmark problems.
When it comes to postprocessing, the reconstruction of the three-dimensional stress field from a beam model is a problem seldom studied. The work of Cesnik [32] and more recently Volovoi [33] tackles this problem by means of a variational asymptotic beam-section analysis which allows recovering of a three dimensional stress state in a postprocessing stage, but since it is based in a Vlasov beam model, its accuracy is limited. In the case of GBT, stress recovery has generally been kept at the most basic level of elasto-kinematic relations which, by the very nature of the GBT kinematics, paint an incomplete picture of the stress profiles. Al-ready in Schardt’s original work, stress reconstruction was limited to normal stresses in the direction of the axis of the beam by means of a classical constitutive relation. In [34], as a part of a buckling analysis of GBT members with non-standard support conditions, the limitations of the elasto-kinematic approach to stress calcula-tion were put in evidence and the recovery of the sole shear stress component along the section midline from axial stress equilibrium was performed. In Chapter 3 of this thesis, following the stress
recovery procedure recently proposed in [35], a procedure for a pos-teriori reconstruction of three-dimensional stresses in the finite el-ement analysis of GBT is presented. The reconstruction is based on the enforcement of the pointwise three-dimensional equilibrium equations over the beam, interpreted as an assembly of thick plates, and allows recovering a fully three-dimensional stress profile with-out the need for corrections to meet the equilibrium boundary con-ditions on bottom/top wall faces. To improve the approximation of stresses over the finite element, a superconvergent patch-based procedure called Recovery by Compatibility in Patches (RCP) is used. This procedure, originally developed for plates [36]-[43], is here suitably extended to the GBT and permits the reconstruction of a stress profile with an accuracy similar to that of a brick finite element model while carrying a computational cost that is several orders of magnitude lower.
Regardless of the abundant literature on the subject, compar-atively little effort has been devoted towards a proper theoretical study of the constitutive relation and its correspondence to the in-ternal constraints of the kinematic model. Given that the GBT essentially considers the cross-section as an assembly of plates, the correct constitutive law to be used for the beam model may not be immediately clear. In the isotropic case, the approach commonly taken in the literature is the adoption of two distinct constitutive laws for the membrane and bending parts of the problem. In the first part of Chapter 4, this issue is addressed and the conventional
approach is presented. On the other hand, in the case of anisotropic thin-walled beams, the Vlasov model was used by Bauld [44] to study fiber-reinforced composite members of open section under different loading conditions. Their model neglected shear deforma-tion, but later efforts by Wu [45], Chandra [46], Kim [47] and others removed this constraint and extended the Vlasov theory to many different types of composites, including laminated profiles. On the GBT front, Silvestre and Camotim [48] were the first to put forward a GBT formulation to account for orthotropic members, studying different cases of lamination and their influence on the GBT equa-tions. Since then, vibration behaviour of arbitrarily loaded lipped channel columns and beams with cross-ply orthotropy has been presented in [49]. The linear and buckling (local and global) be-haviours of FRP composite thin-walled members, with special em-phasis for the effects of material couplings, have been extensively analysed in [50] and [51]. Regardless of the beam model used, most of the approaches available in the literature rely on satisfying inter-nal kinematic constraints by means of a certain condensation of the constitutive matrix, which poses the problem of the correct choice of condensation and defines very different methods of analysis de-pending of the material being used: either different membrane and bending constitutive matrices for the isotropic case or a condensed matrix for the orthotropic material. The second part of Chapter 4 addresses this problem by proposing an alternative writing of the constitutive relations. The kinematic constraints are mirrored onto
the stress profile and a single constitutive matrix is obtained which is correct for both the isotropic and orthotropic cases. In the case of a laminated material, an analogous procedure is presented in which the stress profile of the generic layer is assumed so as to satisfy the kinematic constraints of the laminated plate forming the generic wall. Numerical examples are used to show the performance of this approach with both isotropic, orthotropic and laminated materials. The way in which shear is introduced into the GBT kinematics in Chapter 2 results in a poor description of the membrane shear strain. One way of enriching this description of shear is using a correction factor to adjust for the cross-sectional area that is in fact resistant to shear in a manner analogous to that of the Tim-oshenko beam. As it is well known, these correction factors may not be trivial to determine, especially in the case of an orthotropic material [52]. The final part of Chapter 4 is a discussion on these shear correction factors and a proposal to determine them in the framework of the GBT. Taking advantage of the recovery procedure presented in the third chapter, the elasto-kinematic shear stress is corrected by comparing its energy contribution to that of the recon-structed component. The correction factor thus obtained is used to improve the GBT solution. The results of this approach are shown in numerical examples.
An overview of the mechanics of
thin-walled members
Abstract
In this chapter, cold-formed and composite profiles, which are the main thin-walled beam products of interest of this work, are presented. A brief descrip-tion of their use, history, and fabricadescrip-tion methods is outlined along with the peculiarities of these members. Some of the most important analysis tools for the modelling of these members are described, with special attention being paid to beam models. Specifically, the theory of Capurso and the original GBT formulation. The finite strip method is also briefly presented as a numerical option commonly used in this kind of analysis.
In the most elemental texts on structural mechanics, the term “beam” is used to refer to a structural element that can treated as one-dimensional. The dominant dimension in a beam, referred to as its axis, is the only range of variation for deformations and forces that are defined for a certain cross-section normal to the
t
t << lw
Figure 1.1: Example of a thin-walled cross-section.
axis. Geometrically speaking, this kind of definition points to what is called a beam of “compact” section: a structural member in which all the dimensions of the cross section are similar and in themselves of a smaller order of magnitude than the axial length. Thin-walled beams are a subset of one-dimensional structural elements in which there are three distinctive dimensions instead of two, all of different orders of magnitude. The cross-section is thus understood as a series of segments, with each of them having a dominant dimension in the cross-section plane, as is shown in Fig. 1.1. Each of these segments if referred to as a “wall”, thus the wall thickness is small when compared with its length, which itself small compared with the beam’s length. This geometrical particularity is the basis for the characteristic behaviour of thin-walled beams, whose thorough description is the purpose of this work.
The use of thin-walled members in construction engineering dates back to the 19th century. Indeed, the first I-beams were manufac-tured in France in 1849 and the first steel-frame skyscrapers were built in the U.S. in the 1880s. Cold-formed steel specifically, a prod-uct widely used today for thin-walled beams, also dates back to the mid-19th century but for a long time the scarce knowledge about its mechanical behaviour made its regular and extensive application in construction impractical. Indeed, even after the first standards for structural steel were adopted in the early 20th century, the ge-ometrical particularities of cold-formed members were such that the construction standards could not be readily applied. For this reason, the first widespread applications of cold-formed profiles in construction date from after the Second World War. However, after such slow beginnings, the usage of cold-formed structural elements has increased at a rapid pace and has extended to many different fields of engineering, not only in civil works, but also in aeronau-tical, aerospace, and turbomachinery industries. These profiles are in many cases more economical to produce than other structural members with similar performance, and at the same time can weigh considerably less than the alternatives. Additionally, cold-formed profiles provide significantly more flexibility to construction design, since they can be easily worked into many possible shapes, with each particular geometry offering significant differences in strength characteristics. In this work, attention will be focused on two broad categories of thin-walled beams: cold-formed profiles and composite
profiles. Each of these groups has it own manufacturing process, range of applications, and mechanical behaviour, as we shall see below.
1.1
Cold-formed profiles
One of the oldest and most common method of producing thin-walled beams is cold-forming. This term refers to a range of manu-facturing processes in which a metal sheet is worked by stamping, rolling, pressing, or bending into a usable structural member. The name derives from the forming process being performed at room temperature, as opposed to hot-rolled members, which are heated before forming. Since pressing or rolling a metal sheet at room tem-perature is a relatively easy industrial process, cold-formed profiles can be easily prefabricated and mass-produced. It is from this fact that many of the advantages of these profiles derive. Fig. 1.2 shows some of the many forms in which cold-formed profiles can be pro-duced for commercial use.
The first usage of cold-formed profiles can be traced back to the 1850’s England, but it was not until the American Iron and Steel Institute (AISI) published the first specification for the design of cold-formed profiles in 1946, based in the work conducted by George Winter at Cornell University, that widespread applications of this technology became conveniently available to engineers. From those initial applications to construction, cold-formed profiles have been increasingly used over time for a variety of purposes such framing in
Figure 1.2: Some cold-formed sections.
(a) Building framing using cold-formed profiles
(b) Some types of cold-formed panels
Figure 1.3: Applications of cold-formed structural elements
building construction (Fig. 1.3a), or facesheets for sandwich panels (Fig. 1.3b)
Production process
Cold-formed profiles may be manufactured using two different meth-ods: roll-forming of brake pressing. Roll-forming is a continuous process that shapes a coiled sheet of metal into single profiles. This process is typically performed by means of a series of pairs of rolls
COLD FORM STEEL ROLE
MACHINE
PLAN CUT THE
STUD TO SIZE PUNCHING & DENTING STUD FOR PANEL ASSEMBLY COLD FORM STEEL ROLE ELEVATION
Figure 1.4: Diagram of the roll-forming process.
which rotate in opposite directions drawing the sheet through and slightly changing the shape of the sheet to reach the final config-uration by successive deformations. A diagram of he process is shown in Fig. 1.4. Depending on the complexity of the section desired, more or less rolls will be needed in the process, but in gen-eral roll-forming is a rapid method of producing standard profiles. Given that the manufacturing of different sections will require dif-ferent set of rolls, the process is most economically convenient for mass-production.
Brake-pressing is an alternative procedure useful when more complex sections of cold-formed profiles are needed. The section in this case is manufactured by pressing sheets of metal, forming one or two bends at a time. While this method offers more versatil-ity in the sections that can be formed, it is comparatively crude and slow. Additionally, the length of the profile is limited to the width of the pressing machine, which seldom exceeds 6 meters. Since each finished profile results from a single sheet being folded at a time, this method is not as well suited for mass production as cold
rolling. In any case, the choice of method to use will depend on the needs of the engineer, mainly complexity of the section and required quantity.
1.2
Composite profiles
During the rise of the usage of thin-walled beams for structural applications in the 20th century, the vast majority of these ele-ments were made of cold-formed steel or aluminium. Since these are isotropic materials, for a long time there was no real incentive to conduct research in the field of anisotropic thin-walled beams. The Second World War however, brought about the rise of aero-nautical engineering, which represented a major inflection point in the history of thin-walled structures. In aeronautics, the combina-tion of strength and stiffness with lightness is a focal point for the design of any particular structure. Along with this basic need, a material that is unlikely to completely break under stress, that is a material in which cracks are less easily spread, is of vital impor-tance. Structural elements in aeronautics would also be exposed to extreme conditions of heat and corrosion, and would be needed to be easily manufactured in very complex shapes. For these tasks, metallic thin-walled profiles were ill-fit, so the solution came in the form of composites.
Composite materials, the macroscopic union of two or more ma-terials to produce a third one with desirable properties present in neither of the constituents alone, are in themselves very old.
Ply-wood, the union of different wood sheets at different angles has been know for millennia. Concrete, made of cement in which a certain granular material is embedded, dates from the times of the Ancient Romans; and reinforced concrete, in which steel and concrete are joined to act as a single unit, has been around since the mid 19th century with the appearance of the skyscraper. However, none of these well known materials could satisfy the needs of the aeronau-tics industry, which required thin-walled elements. In this context, a common solution offered is the fiber-reinforced composite, which consists of high strength fibers embedded in a matrix material. Commonly, these fiber-reinforced materials are made in the form of thin layers called lamina. A specific profile may then be formed by stacking the layers (each one independently oriented with respect to the resulting profile) to achieve the desired properties. In recent years, this kind of material has progressively seen its production prices drop until a tipping point was reached in which its applica-tion to the civil industry became economically feasible, mainly in applications such as offshore structures and chemical plants.
Production process
Generally speaking, producing composites involves mixing the ma-trix and fiber materials, binding them together by means of heat, or a chemical reaction, into a single unified product. Depending on the materials used, the final shape desired, and the application details, one of the following methods is used:
Hand Laminating and Autoclave These processes consist on depositing successive layers of material on top of each other, either by hand or robotically. Hand laminating is usually achieved using a male pattern from which a female mold is construed. A gel coat is applied to the male pattern and sev-eral layers of reinforcement are applied alternating with layers of matrix resin, as shown in Fig. 1.5a. This process is com-monly used to make Glass-Fiber-Reinforced-Polymer (GFRP) elements in low-scale production of boats and general pro-totyping. Autoclave processing, schematically shown in Fig. 1.5b also involves the stacking of successive layers, but in this case the layers are made of both the matrix and fiber material that have been previously combined into a semi-finished form called prepreg. After stacking these layers the resulting lam-inated profile is placed inside an autoclave, a pressure vessel that binds the element together at high temperatures. The au-toclave process is used in the aerospace industry and produces components of a very high quality, but it takes great amounts of time and resources.
Filament Winding and fiber placing Both of these processes
similarly revolve around applying reinforcements around a man-drel. In the case of filament winding, fibers are wound over a rotating mandrel (see Fig. 1.6a) following a specific pattern of fiber orientation that gives the finished product its final mechanical properties. In the case of fiber placing,
reinforce-(a) Hand laminating
(b) Autoclave process
Figure 1.5: Schematic view of the hand laminating and autoclave processes
(a) Filament winding
(b) Fiber placing
Figure 1.6: Schematic view of the filament winding and fiber placing processes
ments are applied to the surface of the mandrel in the form of wrapping strips, as shown in Fig. 1.6b, held in place by ap-plying high pressure and temperature. Both of these processes are used to create surfaces of revolution and result in elements that perform very well under high pressures, thus making them ideal for tanks and pipes.
Pultrusion This process is a combination of pulling and extruding (hence its name) and consists of fibers impregnated with resin that are pulled through a stationary heated die (see Fig. 1.7),
Figure 1.7: Schematic view of the pultrusion process
which solidifies the resin. Upon exiting the die, the structural component is pulled and cut to length in a similar way to cold-rolled metal profiles. This procedure allows the production of composite beams at a low cost and with a high production rate, thus making pultruded beams a good component to use in the construction industry.
Liquid Molding In this family of processes the fibers are pack-aged into a pre-form having the configuration of the final part. The preform is then placed inside a processing mold and resin is infused at high pressure to wet the fibers and fill any ex-isting cavity, usually catalysts and heat are used to help the process. After the resin has been infused, the part is demolded and ready to be used. This process allows for relatively low cost and high production rates, in a way similar to pultrusion. Fig. 1.8 shows a schematic view of this process.
Figure 1.8: Schematic view of the liquid molding process
1.3
Theoretical framework
It is a well known fact that in thin-walled beams torsional loads produce also normal strains and stresses, and longitudinal or trans-verse loads induce torsion. This coupling of two different sets of kinematics stems of the fact that the cross-section of a thin-walled beam cannot be assumed to remain plane as it would be in compact-section beams, as the cross-compact-section experiences out-of-plane warp-ing in response to torsion. In addition to this, the distortion of the cross-section in its own plane is an aspect that can be critical to structural design depending on the application, as thin-walled cross-sections are essentially folded plates. Moreover, thin-walled profiles are particularly prone to instability phenomena which significantly differ from structural members with compact section. Specifically, thin-walled beams can not only suffer global instability, in which a certain deformation is experienced along the member’s longitudinal axis (see Fig. 1.9a) leaving the cross-section undeformed in its own plane, but can also experience distorsional instability characterized by a deformation of the whole cross-section over a certain axial
(a) Global buckling
(b) Distortional buckling (c) Local buckling
Figure 1.9: Buckling modes
length (Fig. 1.9b) and even local instability, in which deformation may be limited to the single walls of the cross-section (Fig. 1.9c).
In light of the particularities explained above, it can be said that even if the manufacturing process of thin-walled profiles is relatively simple, their structural design is far from it. Indeed, any serious attempt to model the mechanics of thin-walled beams must take into account not only cross-section warping, but also shear deformation, cross-section distortion and local effects, while at the same time allowing for the study of many different loading cases and a wide assortment of boundary conditions. The first significant improvement from the classical Euler-Bernoulli beam into the field of thin-walled beams was the work of Vlasov [5] in 1940. The model of Vlasov, also known as the Theory of the Sectorial Area, deals with out-of-plane warping of the cross-section and revolves around a warping function that essentially characterizes the kinematics of non-uniform torsion along the beam axis. The work of Vlasov is still widely used as a standard today for the treatment of thin-walled beams and has been enriched by the efforts of many authors, since the early contributions of Timoshenko and Gere, which described
the stability of thin-walled beams under diverse loading conditions [7] or Murray [53] which applied the theory to standard design practices, to the more recent work on the stability of thin-walled profiles by Wilson et al. [10] and Kim et al [9]. However, the model of Vlasov has several limitations at its core: the cross-section is considered to be perfectly rigid in its own plane, the shear strains in the middle surface of the wall are neglected, and the transverse normal stresses in the walls are ignored along with normal stresses tangent to the midsurface of the wall. These limitations may be more or less relevant depending on the specific case considered, but it is clear that models based on the assumptions of Vlasov are an inherently incomplete description of the mechanics of thin-walled beams.
1.3.1 The theory of Capurso
A first generalization of the well known model of Vlasov is the Ca-purso Beam Theory. In the 1960s, Michele CaCa-purso [12] introduced a novel model for thin-walled beams based on not just one, but instead a series of warping functions over the cross-section. In his seminal work, Capurso added to the six classical degrees of freedom of a Vlasov beam (axial displacement, displacements and rotations over the two main axes of inertia, and non-uniform torsion) infi-nite degrees of freedom that describe the many possible shapes in which the cross-section could warp. In essence, this constitutes a generalization of the cross-section’s degree of freedom concept. If
we consider a generic thin-walled section, such as the one in Fig. 1.10 we can write the displacement field in the x, y, z coordinate system as: dx(s, z) = u(z) − ϕz(z) (y(s) − yC) , (1.1) dy(s, z) = v(z) − ϕz(z) (x(s) − xC) , (1.2) dz(s, z) = w(z) − ϕy(z)x(s) + ϕx(z)y(s) + n X i=1 ϕi(z)φi(s), (1.3) s G x y xC yC C
Figure 1.10: Capurso theory: global and local reference systems on a generic thin-walled cross-section.
where dx, dy, dz are the displacements of a generic point on the
cross-section in the x, y, z directions, functions u(z), v(z), w(z) are the displacements in the x, y, z directions of the shear center, which is located in the coordinates xC, yC, and ϕx, ϕy, ϕz are the rotations
of the cross-section around axes x, y, z. The generic φi(s) is a
This description not only contains the kinematic of Vlasov, but also does away with the assumption of null shear strains in the middle surface of the wall.
It is of particular importance to the present work that Capurso developed a modal theory, since each new warping function added to the kinematic description is weighed depending on the way the beam is loaded. The appeal of this modal approach will be better understood later on when more sophisticated model are presented, but it should be noticed that this is, to the knowledge of the author, the first instance of a modal description of the mechanics of thin-walled beams beyond the theory of Vlasov.
The model of Capurso remains valid today and in fact has been recently extended to transversely isotropic materials and applied to the analysis of pultruded fiber-reinforced polymer (FRP) thin-walled beams [54][55]. However, as rich as it is, the work of Capurso still lacks some elements that would be desired in a comprehensive model for thin-walled beams. Chief among these missing elements is the cross-section distortion. Without it, in-plane deformations of the cross-section and local phenomena cannot be described, and given the importance of these in the structural performance of thin-walled beams, it is clear that additional effort is needed.
1.3.2 The Generalized Beam Theory
An alternative beam model useful for the analysis of thin-walled beams is the Generalized Beam Theory (GBT) developed by Schardt
[17][18] in the 1980s. The GBT can be viewed as a generalization of the Vlasov theory able to take into account in-plane cross-section deformations while rewriting the beam kinematics in a modal form analogous to that of Capurso. Indeed, the GBT unifies conventional thin-walled beam theories and extends them to include section dis-tortion by treating each particular set of kinematics (bending, tor-sion, distortion, etc.) as merely a special case of a generalized model.
The fundamental idea of the GBT is to consider a thin-walled beam as an “assembly" of thin plates and to assume the displace-ment field of the beam as a linear combination of predefined cross-section deformation modes (which are known beforehand) multi-plied by unknown functions depending on the beam axial coordi-nate, that can be called kinematic parameters or generalized dis-placements. This analytical treatment defines four fundamental deformation modes (so called because they must absolutely be in-cluded in any GBT formulation) to account for extension, bending about the two principal axes and non-uniform torsion. These modes are also referred to in the literature as “rigid-body modes” because they do not involve any distortion of the cross-section. Additional deformation modes may then be used to include distortion and local effects. This unified writing is not only an elegant way of transition from a beam model to a folded plate problem, but is also very con-venient to structural design since all of the classical beam quantities can be recovered directly as special cases of the generalized model.
In the original GBT formulation, the displacement field is de-fined for the generic i-th wall of the cross-section (see Fig. 1.11) as:
dn(n, s, z) = ψ(s)v(z), (1.4)
ds(n, s, z) = [µ(s) − n∂sψ(s)] v(z), (1.5)
dz(n, s, z) = [ϕ(s) − nψ(s)] ∂zv(z), (1.6)
where dn is the displacement orthogonal to the wall midline, ds is
the displacement tangent to the wall midline, dz is the displacement
in the beam axial direction, ψ, µ and ϕ are row matrices collect-ing the assumed cross-section deformation modes (dependcollect-ing only on s) and v is the vector that collects the unknown kinematic pa-rameters (depending only on z). Moreover, ∂s and ∂z denote the
derivative with respect to the s coordinate and to the z coordinate, respectively. In the following, the term natural nodes is used to re-fer to the vertices of the cross-section midline, while internal nodes refers to intermediate points along the wall midline, as shown in Fig. 1.11.
In a similar way to the Capurso theory, the choice of the deforma-tion modes constitutes the core of the GBT. As we shall see shortly, this choice of modes is intrinsically connected to the discretization of the cross-section into natural and internal nodes, since each node considered is the basis for a deformation mode. The procedure by which modes are defined from the cross-section discretization is
re-n s 1 2 … natural nodes internal nodes
Figure 1.11: GBT: natural and internal nodes in a generic thin-walled cross-section.
ferred to as cross-section analysis and it is in fact a separation of variables, since by performing a cross-section analysis the in-plane kinematics of the cross-section is defined. The original GBT for-mulation of Schardt relied on the so-called flexural modes, which could be divided in fundamental flexural modes and local flexural modes.
Fig. 1.12a shows how one fundamental flexural mode can be de-fined for each natural node by assuming along the midline a piece-wise linear (linear on each wall) warping function. This warping function has a unit value at the corresponding natural node and is null for all other natural nodes in the cross-section. Analogous functions are defined for the rest of the natural nodes, so the warp-ing description of the cross-section is complete. It is important to note that, while the enrichment of the warping description in this modal way is similar to that of Capurso, the fundamental modes of
(a) (b) (c) dz dz ˆ ˆ ˆ Shear mode ˆ ˆ ˆ ˆ Flexural mode dz ˆ ˆ ˆ ˆ dz 2 ˆ ˆ ˆ ˆ ˆ dz 2ˆ 2ˆ 1 Ψj j i φi
Figure 1.12: GBT: (a) warping in a fundamental flexural mode, (b) in-plane displacements in a fundamental flexural mode, (c) in-plane displacements in a local flexural mode.
the GBT are different in nature, since their warping description is linear. In addition to this, another key aspect of the fundamental modes is the fact that strain γzs is assumed to be null, thus
follow-ing the assumption of Vlasov. Later on, we shall see that another version of the GBT can be formulated without recurring to this assumption, but for the time being this conditions implies:
ϕ(s) linear, (1.7)
µ(s) = −∂sϕ(s) ⇒ constant, (1.8)
ψ(s) cubic, (1.9)
where the function µ represents the in-plane displacement of the current natural node in the direction of the midline. Clearly, the relation between µ and ϕ is the result of Vlasov’s assumption. With these displacements completely known, the in-plane displacement transversal to the wall midline, denoted by ψ, are easily determined
by considering the cross-section as a planar frame and solving it for ψ with the prescribed values of µ as end conditions, thus restoring compatibility along the walls subjected to cylindrical bending (see Fig. 1.12b). This procedure is a key feature of the GBT and, after being performed for all the modes, allows to recover the kinematics of a Vlasov beam enriched with Nn − 4 distortional modes, being
Nn the number of natural nodes.
Local flexural modes can be defined in a similar way. One local flexural mode is considered for each internal node by assuming null warping, null shear strain γzs along the midline of the cross-section
and piecewise cubic displacement normal to the wall. Thus, these modes enrich the in-plane kinematic description of the cross-section while not contributing to warping. Functions ψ are determined by assuming unit normal displacement at a certain internal node and zero at the other nodes (see Fig. 1.12c) so their value over each wall is again determined by solving the planar frame of the section with the prescribed conditions, i.e. null µ:
ϕ(s) = µ(s) = 0, ψ(s) piecewise cubic. (1.10)
While the number of fundamental modes is limited by the geom-etry of the section (there can be only one per vertex), the number of possible local modes is potentially infinite. By simply refining the discretization of each wall, the description of local phenomena can be enriched as much as needed. This is a significant difference with the models of Vlasov and Capurso, both of which neglect local
effects and distortion.
As said above, the first four fundamental flexural modes corre-spond to the kinematics of a Vlasov beam, with the rest adding distortion and local effects. This, of course, is not immediately ob-vious from the way the modes are defined. To make this clear, and indeed to give each mode a distinct physical meaning, it is neces-sary to transform Eqs. (1.4)-(1.6) through a modal decomposition. In particular, the system can be transformed from its original base to a modal one in which each cross-section deformation mode has a specific meaning. Indeed, it can be shown that, in the modal space, the first four modes correspond to the classical degrees of freedom of a shear-undeformable beam and each additional mode corresponds to section distortion and/or local wall deformation. As an example, in Fig. 1.13 the displacements corresponding to the six fundamental flexural modes of a C-shaped cross-section after such modal decomposition are shown. It can be seen how the classical generalized deformations of a Vlasov beam are recovered, such as axial extension (mode 1), major and minor axis bending (modes 2 and 3), and twisting rotation about the shear centre (mode 4). Modes 5 and 6 are typical GBT higher-order flexural deformations involving section distortion.
1.4
Numerical Modelling
Given the three-dimensional character of the strain field in thin-walled mechanics, a common way to describe these problems is by
Mode 1 Mode 2
in-plane displacements out-of-plane displacements in-plane displacements out-of-plane displacements
Mode 3 Mode 4
in-plane displacements out-of-plane displacements in-plane displacements out-of-plane displacements
Mode 5 Mode 6
in-plane displacements out-of-plane displacements in-plane displacements out-of-plane displacements
Figure 1.13: C cross-section modelled by GBT: in- and out-of-plane displace-ments corresponding to the six fundamental flexural modes after modal de-composition.
means of the Finite Element Method (FEM). A thin-walled beam could be modelled as an assembly of plate elements and thus its complete mechanical behaviour could be determined. However, such an approach would carry an important computational cost, since the FEM solution requires discretisation in every dimension of the problem and therefore considerably more unknowns than a beam-based approach. Moreover, a FEM solution is exclusively three-dimensional, which means that classic one-dimensional quan-tities such as bending moments, normal stress, and shear force (or their kinematic counterparts, like rotations, axial displacement, and shear deflections) cannot be easily identified. This is a significant drawback from the point of view of the structural designer, who needs to use these quantities to follow engineering standards. En-riched beam models, such as that of Capurso or the GBT, are a useful alternative to fully three-dimensional FEM models in this sense. In the following chapters, a numerical implementation of the GBT will be presented as the preferred model for thin-walled beams. However, for the sake of completeness, a numerical ap-proach referred to as the Finite Strip Method (FSM), commonly used to model thin-walled profiles, will be briefly described here. This method is born out of the need to compromise between the kinematic richness of a plate FEM approach and the computational economy of a beam model, and has been proven to give satisfactory results for a variety of problems.
(a) FEM (b) FSM
Figure 1.14: Comparison of the level of discretization used for the same geom-etry using finite elements and finite strips
1.4.1 The Finite Strip Method
The FSM is a semi-analytical procedure that stands in between the classical Rayleigh-Ritz method and a FEM solution. At its base there is a separation of variables, in a similar way to that of Kantorovich [56], between the axial and cross-sectional directions. From this separation, two different sets of functions can be used to approximate the variables in the corresponding directions.
In the FSM, a beam is divided into longitudinal strips (hence the name of the method) adjacent among themselves, as shown in Fig. 1.14b. Two different sets of approximation functions are then chosen: one along the axial direction and another over the cross-section. Since this sort of discretisation creates significantly less un-knowns than a FEM approach (see the equivalent FEM mesh in Fig. 1.14a) the FSM method is able to describe the three-dimensional strains of the problem with reduced computational effort.
choice of the approximation functions. The classical FSM [57]called for using simple polynomials, such as Hermite functions, as shape functions over the cross-section along with continuously differen-tiable smooth trigonometrical or hyperbolic functions over the axial direction. More recent versions of the method use different func-tions, such as splines, to overcome the difficulties of the interpola-tion, but the principle of separation of variables remains. Whatever the shape functions may be, the general form of the displacements can be written as:
w(s, z) =
n
X
i=1
φi(s)ϕi(z), (1.11)
where φi(s) represents the i-th polynomial shape function over the
cross-section. This part of the interpolation must represent a state of constant strain over the cross-section, so as to guarantee conver-gence as the mesh is refined. The ϕi(z) part of the interpolation is
the longitudinal series of trigonometrical, hyperbolic or spline func-tions. This series must evidently satisfy the end conditions imposed on the beam. The choice of these functions is of vital importance in FSM analysis, since no refinement over the strip length is possible. This means that the FSM is susceptible to the specific load and end conditions, with the risk of not being applicable in some cases.
Since each strip is considered to be under a plane stress state and based in the Kirchhoff thin plate theory, the FSM allows to take into account not only the enriched warping description present in the Capurso theory, but also the distortion of the cross-section
and local effects. However, the family of solutions obtained is diffi-cult to classify into meaningful modes, so most current applications of the FSM introduce specific mechanical criteria to separate the displacement field into distinct subspaces. These mechanical cri-teria, directly motivated by the GBT [58], form the basis of the so-called Constrained Finite Strip Method (cFSM) and generally produce satisfactory results for the stability analysis of thin-walled beams. The cFSM has been successfully used by Schafer [59] to an-alyze cold-formed steel members and has been used to differentiate global, distortional, and local buckling modes on open cross-section thin-walled beams [60]. However, even when using different longi-tudinal shape functions, some seemingly insurmountable obstacles remain in the way, mainly:
• Arbitrary boundary conditions are not possible to impose over the member. In particular, longitudinal elastic restraints (much common in cold-formed members applications) are not allowed by the formulation.
• Even when applying the cFSM methodology to identify dis-tinct modes, these don’t have a clear mechanical meaning. This means the classical degrees of freedom and stress resul-tants are not recovered exactly, which thwarts any attempt to use current standards for structural design.
The limitations of the FSM are such that it would be impractical to use for a general study of the mechanics of thin-walled profiles. A
methodology that brought together the versatility of the Capurso theory with the kinematic richness of the FSM is needed. This research work is bent on developing such a methodology based on the GBT.
A Generalized Beam Theory with
shear deformation
Abstract
In this chapter, a GBT formulation that coherently accounts for shear deforma-tion is presented. The new formuladeforma-tion, which preserves the general format of the original GBT for flexural modes, introduces the shear deformation along the wall thickness direction besides that along the wall midline, so guaran-teeing a coherent matching between bending and shear strain components of the beam. A reviewed form of the cross-section analysis procedure is devised, based on a unique modal decomposition for both flexural and shear modes. The effectiveness of the proposed approach is illustrated on three benchmark problems.
As shown in Chapter 1, the original formulation of the GBT maintained the Vlasov assumption of null shear deformation of the generic wall. It wasn’t until almost two decades later that the work of Silvestre and Camotim [29]-[30] removed this constraint
by adding a new family of shear modes to the GBT formulation. These modes are based on the inclusion of non-null warping ϕs in
the kinematics together with null in-plane displacement. This leads to the following displacement field (see [49]-[51]):
dn(n, s, z) = ψ(s)v(z), (2.1)
ds(n, s, z) = [µ(s) − n∂sψ(s)] v(z), (2.2)
dz(n, s, z) = [ϕ(s) − nψ(s)] ∂zv(z) + ϕs(s)δ(z), (2.3)
where an additional generalized degree of freedom δ(z) is added to dz along with a new warping function ϕs(s). As has been noted in
[51], the above warping kinematics can be recast in the conventional GBT form presented in Chapter 1 by assuming
ψ(s) = µ(s) = 0, ϕ(s) 6= 0. (2.4)
In other words, the shear modes introduced in this manner are in fact analogous to the conventional GBT modes with the difference being the imposition of null in-place displacements. However, it is important to notice that these modes can be defined for both natural and internal nodes, which means that ϕs can be used to
describe a non-linear warping displacement along the wall midline. Indeed, shear modes defined in this manner are formally analogous to those proposed in the theory of Capurso, presented in Chapter 1. Moreover, it is worth noticing that the warping description remains constant along the wall thickness.
Adding shear deformability in the manner presented above clearly removes the Vlasov constraint of null shear strain γzs along the
midline of the cross-section. However, given that the warping de-scription is constant along the wall thickness, shear strain between the direction of the beam axis and the normal to the wall midline γzn results to be null. In terms of the global beam behaviour, this
translates into a lack of coherence between the bending and shear strain components of the beam. This problem is illustrated in Fig. 2.1, where a schematic view of the bending and shear strain compo-nents of a beam is shown for the case of a Timoshenko beam and for the conventional GBT model. It becomes clear from this represen-tation that due to the absence of a non-null γzn the shear strain in
a classical shear deformable GBT is inconsistent through the wall thickness with the bending strain component of the beam. This mismatch engenders several theoretical and practical drawbacks:
• An ad hoc modal decomposition procedure must be used for shear modes, completely different from the one used for the flexural counterpart.
• There are two generalized strain components associated to each shear mode, instead of a single one as would be reasonable to expect.
• Classical shear deformable beam theories are not recovered exactly.
flange
flange web
dz bending shear shear
gzs ≠ 0 gzn = 0 gzs ≠ 0 gzn ≠ 0 (d) (c) (b) (a)
Figure 2.1: Sketch of the main difference between classical shear deformable GBT and the present formulation: (a) undeformed elementary beam, (b) bend-ing strain component, (c) correspondbend-ing shear strain component in the classi-cal GBT, (d) corresponding shear strain component in the Timoshenko beam theory and in the present theory.
since one of the main advantages of the GBT is precisely that clas-sical beam theories can be recovered as special cases. The conven-tional approach to including shear deformability in the GBT is such that, even after modal decomposition, it is not possible to obtain the typical deformation components of classical theories: bending deflections are not clearly distinguishable from deflections due to shearing strains and, in consequence, pure cross-section flexural rotations do not result as natural kinematic parameters. Further-more, these problems have definite consequences for a GBT numer-ical implementation: a GBT-based finite element formulated with this theory would not be efficient from the computational point of view and could not be combined with standard elements since the cross-section rotation is not a degree of freedom. For this same
reason, corotational approaches to solve non-linear problems can-not be easily applied in this kind of GBT framework. All of this severely limits the attractiveness of the shear-deformable GBT as has been traditionally presented.
2.1
New kinematics
To properly resolve the problems presented above, a more com-prehensive way of introducing shear deformability into the GBT is necessary. In this respect, a new displacement field can be defined for the i-th wall (see Fig. 1.11):
dn(n, s, z) = ψ(s)v(z), (2.5)
ds(n, s, z) = [µ(s) − n∂sψ(s)] v(z), (2.6)
dz(n, s, z) = [ϕ(s) − nψ(s)] [∂zv(z) + δ(z)] + ϕh(s)δh(z). (2.7)
In line with traditional GBT formulations, the unknown gener-alized displacements are divided between those related to flexural modes v and those corresponding to shear modes. In this case, however, there are two distinct parameters related to shear instead of one. The δ parameter corresponds to basic shear modes, which are in the same number of flexural (both fundamental and local) modes. The second δh parameter corresponds to additional shear
modes, which may be used to describe non-linear warping. It is eas-ily seen that if δ and δh are disregarded then the new displacement
thus this new formulation contains the original one as a special case. In this GBT formulation, δ is the typical term of shear deformable theories, such as the Timoshenko beam theory, in which longitudi-nal displacement is not defined by the derivative of the transverse displacement v. This term contributes in a different manner from the classical shear deformable GBT shown in Eqs. (2.1)-(2.3), since the term −nψδ is present. Specifically, this formulation allows for a variation of the warping displacement associated to δ along the wall thickness, in the same way the flexural warping does. The ad-ditional δh parameters further enrich the warping description over
the cross-section midline (that is over the s coordinate), but these are constant along the wall thickness n. Indeed, these additional shear parameters carry a contribution that is formally analogous to that present in Eqs. (2.1)-(2.3) and in the theory of Capurso.
2.2
Strains and generalized deformations
From the displacement field defined in Eqs. (2.5)-(2.7), strains can be calculated through the point-wise compatibility equations:
εzz(n, s, z) = [ϕ(s) − nψ(s)]∂z2v(z) + ∂zδ(z) +
+ϕh(s)∂zδh(z), (2.8)
εss(n, s, z) = ∂sµ(s) − n∂s2ψ(s) v(z), (2.9)
γzs(n, s, z) = [µ(s) − 2n∂sψ(s) + ∂sϕ(s)] ∂zv(z) +
+∂sϕh(s)δh(z), (2.10)
γzn(s, z) = −ψ(s)δ(z), (2.11)
εnn = 0 (2.12)
γsn = 0 (2.13)
It is important to note that, differently form the classical shear deformable GBT, the shear strain γzn is different from zero. As it
can be easily realized, the six independent z-fields governing the strain components can be grouped together in a vector e of gener-alized deformation parameters:
eT = h vT ∂
zvT (∂z2v + ∂zδ)T δT δh T ∂zδh T
i
. (2.14)
For the sake of convenience, Eqs. (2.8)-(2.11) can be written in a more compact form by introducing vector ε = h εzz εss γzs γzn iT, resulting in:
ε(n, s, z) = b(n, s)e(z), (2.15)
where the modal matrix
b(n, s) = 0 0 ϕj− n ψj 0 0 ϕhj −n ∂2 sψj 0 0 0 0 0 0 −2n∂sψj 0 ∂sϕj− n∂sψj ∂sϕhj 0 0 0 0 −ψj 0 0 (2.16)
collects the cross-section deformations for each mode. More clearly, the generalized deformations can be rewritten as
e = Du, (2.17) where u = v δ δh , D = Im ⊗ L, L = 1 0 0 ∂z 0 0 ∂z2 ∂z 0 0 1 0 0 0 1 0 0 ∂z , (2.18)
In this writing, Im is the m-order unit matrix, m is the number
of the deformation modes, and symbol ⊗ denotes the Kronecker product. The statement in Eq. (2.17) means the differential opera-tor D can be interpreted as the compatibility operaopera-tor of the beam model.
Since it will be convenient later on, strains Eqs. (2.8)-(2.11) can be separated into a “membrane" part, not depending on n and denoted by superscript (M), and a “bending" part, proportional to n and denoted by superscript (B), as follows:
εzz(n, s, z) = ε(M )zz (s, z) + nˆε(B)zz (s, z), (2.19)
εss(n, s, z) = ε(M )ss (s, z) + nˆε(B)ss (s, z), (2.20)
γzs(n, s, z) = γzs(M )(s, z) + nˆγzs(B)(s, z), (2.21)
where ε(M )zz = ϕ(s)∂2 zv(z) + ∂zδ(z) + ϕh(s)δh(z), (2.23) ˆ ε(B)zz = −ψ(s)∂2 zv(z) + ∂zδ(z) , (2.24) ε(M )ss = ∂sµ(s)v(z), (2.25) ˆ ε(B)ss = −∂s2ψ(s)v(z), (2.26) γzs(M ) = [µ(s) + ∂sϕ(s)] ∂zv(z) + ∂sϕ(s)δ(z) + ∂sϕh(s)δh(z), (2.27) ˆ γzs(B) = −2∂sψ(s)∂zv(z) − ∂sψ(s)δ(z) (2.28) (2.29)
2.3
Generalized stresses
Generalized stresses s are the dual quantities of the generalized deformations defined in Eq. (2.14) and they can be determined according to the following work equivalence condition:
Z s Z n σTε dn ds = sTe, (2.30) where εT=[ε zz εss γzs γzn] T and σT=[σzz σss τzs τzn] T. It is
im-portant to notice that e contains cross-section rigid-body motions, so the s generalized stresses associated to these are meaningless.
Additionally, the equilibrium equations of the beam model can be written by means of the principle of virtual work. That is:
D∗s = q, (2.31) with the equilibrium operator:
D∗ = Im⊗ L∗, (2.32)
being L∗ the differential operator adjoint to L and q the vector
that collects the generalized loads, work-conjugate of the kinematic parameters u.
2.4
Cross section stiffness matrix
In this chapter, an elastic isotropic material will be considered to write the cross-section stiffness matrix. Later on, a more general treatment of the constitutive relation will be introduced and the expressions obtained below will be generalized to the case of com-posite materials. However, let us begin with the simplest case of an elastic isotropic material. It may not be evident upon first inspec-tion, but the kinematics defined in Eqs. (2.5)-(2.7) lead to internal constraints that render the beam model overstiff, which is a well know fact reported in the literature. Details on these constraints in the case of the GBT, their consequences, and the way in which they can be satisfied will be given in Chapters 3 and 4. For now, let it suffice to know that some relaxing of the stiffness is needed to balance the effect of internal constraints. The usual way of doing this is an adjustment of the constitutive relation. In this sense, the
membrane-bending separation introduced in the previous section can be exploited by using different constitutive relations for each part: σ = C(M )ε(M ) + C(B)ε(B). (2.33) where C(M ) = E 0 0 0 0 E 0 0 0 0 G 0 0 0 0 G , C(B) = E 1−ν2 1−ννE2 0 0 νE 1−ν2 1−νE2 0 0 0 0 G 0 0 0 0 G , (2.34)
being E the Young modulus, ν the Poisson coefficient, and G the shear modulus. Thus, an uncoupled constitutive relation, typical of a beam solution, is assumed for the membrane part of the problem. For the bending part, a coupled constitutive relation is assumed, as is done for plates. From this assumption, the cross-section stiffness matrix C, that is the constitutive relation of the beam model, can be determined by writing a work-equivalence condition:
Z s Z n h ε(M )+ nˆε(B)iThσ(M ) + nˆσ(B)i dn ds = eTCe. (2.35) Evaluating the above equation gives us a cross-section stiffness matrix, of the form:
